Preface |
First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves / 1.: |
The Riemann Sphere / 1.1: |
Complex Manifolds / 1.2: |
Rational Functions / 1.3: |
Luroth's Theorem / 1.4: |
Automorphisms of P[superscript 1] / 1.5: |
Spherical Geometry / 1.6: |
Finite Subgroups and the Platonic Solids / 1.7: |
Automorphisms of the Half-Plane / 1.8: |
Hyperbolic Geometry / 1.9: |
Projective Curves / 1.10: |
Covering Surfaces / 1.11: |
Scissors and Paste / 1.12: |
Algebraic Functions / 1.13: |
Examples / 1.14: |
More on Uniformization / 1.15: |
Compact Manifolds as Curves: Finale / 1.16: |
Elliptic Integrals and Functions / 2.: |
Elliptic Integrals: Where They Come From / 2.1: |
The Incomplete Integrals Reduced to Normal Form / 2.2: |
The Complete Integrals: Landen, Gauss, and the Arithmetic-Geometric Mean / 2.3: |
The Complete Elliptic Integrals: Legendre's Relation / 2.4: |
The Discovery of Gauss and Abel / 2.5: |
Periods in General / 2.6: |
Elliptic Functions in General / 2.7: |
The and-Function / 2.8: |
Elliptic Integrals, Complete and Incomplete / 2.9: |
Two Mechanical Applications / 2.10: |
The Projective Cubic / 2.11: |
The Problem of Inversion / 2.12: |
The Function Field / 2.13: |
Addition on the Cubic / 2.14: |
Abel's Theorem / 2.15: |
Jacobian Functions: Reprise / 2.16: |
Covering Tori / 2.17: |
Finale: Higher Genus / 2.18: |
Theta Functions / 3.: |
Jacobi's Theta Functions / 3.1: |
Some Identities / 3.2: |
The Jacobi and Weierstrass Connections / 3.3: |
Projective Embedding of Tori / 3.4: |
Products / 3.5: |
Sums of Two Squares / 3.6: |
Sums of Four Squares / 3.7: |
Euler's Identities: Partitio Numerorum / 3.8: |
Jacobi's and Higher Substitutions / 3.9: |
Quadratic Reciprocity / 3.10: |
Ramanujan's Continued Fractions / 3.11: |
Modular Groups and Modular Functions / 4.: |
The Modular Group of First Level / 4.1: |
The Modular Group of Second Level / 4.2: |
Fundamental Cells / 4.3: |
Generating the Groups / 4.4: |
Gauss on Quadratic Forms / 4.5: |
The Group of Anharmonic Ratios / 4.6: |
Modular Forms / 4.7: |
Eisenstein Sums / 4.8: |
Absolute Invariants / 4.9: |
Triangle Functions / 4.10: |
The Modular Equation of Level 2 / 4.11: |
Landen's Transformation / 4.12: |
Modular Equations of Higher Level / 4.13: |
Jacobi's Modular Equation / 4.14: |
Jacobi and Legendre's Derivation: Level 5 / 4.15: |
Arithmetic Subgroups: Overview / 4.16: |
Ikosaeder and the Quintic / 5.: |
Solvability of Equations of Degree [less than] 4 / 5.1: |
Galois Groups Revisited / 5.2: |
The Galois Group of Level 5 / 5.3: |
An Element of Degree 5 / 5.4: |
Hermite on the Depressed Equation / 5.5: |
Hermite on the Quintic / 5.6: |
A Geometric View / 5.7: |
Imaginary Quadratic Number Fields / 6.: |
Algebraic Numbers / 6.1: |
Primes and Ideal Numbers / 6.2: |
Class Invariants and Kronecker's Jugendtraum / 6.3: |
Application of the Modular Equation / 6.4: |
The Class Polynomial / 6.5: |
Class Invariants at a Prime Level / 6.6: |
Irreducibility of the Class Polynomial / 6.7: |
Class Field and Galois Group / 6.8: |
Computation of the Class Invariants / 6.9: |
Arithmetic of Elliptic Curves / 7: |
Arithmetic of the Projective Line / 7.1: |
Cubics: The Mordell--Weil Theorem / 7.2: |
Proof of the Mordell--Weil Theorem / 7.3: |
References |
Index |
Preface |
First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves / 1.: |
The Riemann Sphere / 1.1: |